There's not much difference. They are almost interchangeable most of the time.
"Common" simply means "frequent."
"Normal" means "expected," It often carries the idea of "correct" and "proper."
For example, in the United States, it is common to put ketchup on hamburgers and mustard on hot dogs. Some people put ketchup on hot dogs; that's less common, but it is normal. But, if someone put chocolate syrup on a hot dog, that would be very strange, and people would say "that's not normal."